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Top colleges have pledged to become more socioeconomically diverse, but the admissions edge many give to children of alumni may make that goal harder to achieve.

At the University of Notre Dame, the University of Virginia and Georgetown University, the admission rate for legacies is about double the rate for the overall applicant pool, according to data from the schools. At Princeton University, legacies are admitted at four times the general rate, or roughly 30% compared with about 7% overall over the past five years, the school says.

Legacy applicants at Harvard University were five times as likely to be admitted as non-legacies, according to an analysis of admissions data from 2010 through 2015. The numbers—33.6% for legacies and 5.9% for those without parental ties—were submitted in a June court filing for a case claiming Asian students are being discriminated against in the name of greater diversity at the school.

All of those schools have signed on to or plan to join the American Talent Initiative, a Bloomberg Philanthropies-backed effort to enroll 50,000 more low- and moderate-income students by 2025.

Concerns over the legacy advantage reflect broader unease about competing priorities in admissions. Diversity initiatives have led to complaints by white students that minority students have a leg up. Meanwhile, highly qualified Asian students say they should get more slots based on academics. Both say long-standing traditions like legacy admissions soak up coveted spots.

Advocates for considering legacy status argue that favoring the children—and, in some cases, grandchildren—of graduates helps maintain an engaged and generous alumni base and lets students serve as ambassadors to new campus arrivals.

Cornell University President Martha E. Pollack has said legacy admissions help perpetuate “a Cornell family that goes on for generations.” In an interview with the student newspaper in May, she said the practice isn’t about giving preference or an advantage to legacies, but such a designation is one of many “balancing factors.”

Colleges should be honest about what they are doing. I don't see how Cornell can use legacy status as a "balancing factor" without giving legacies some preference.